The most important of Cayley's work was in developing the algebra of matrices, in non-euclidean geometry and n-dimensional geometry.

Arthur's father Henry Cayley, although from a family who had lived for many generations in Yorkshire, England, lived in St Petersburg, Russia. It was in St Petersburg that Arthur spent the first eight years of his childhood before his parents returned to England and settled near London. Arthur showed great skill in numerical calculations at school and, after he moved to King's College School in 1835, his aptitude for advanced mathematics became apparent. His mathematics teacher advised that Arthur be encouraged to pursue his studies in this area rather than follow his father's wishes to enter the family business as merchants.

In 1838 Arthur began his studies at Trinity College, Cambridge from where he graduated in 1842. While still an undergraduate he had three papers published in the newly founded Cambridge Mathematical Journal edited by Duncan Gregory. Cayley graduated as Senior Wrangler and won the first Smith's prize. For four years he taught at Cambridge having won a Fellowship and, during this period, he published 28 papers in the Cambridge Mathematical Journal .
A Cambridge fellowship had a limited tenure so Cayley had to find a profession. He chose law and was admitted to the bar in 1849. He spent 14 years as a lawyer but Cayley, although very skilled in conveyancing (his legal speciality), always considered it as a means to make money so that he could pursue mathematics.

While still training to be a lawyer Cayley went to Dublin to hear Hamilton lecture on quaternions. He sat next to Salmon during these lectures and the two were to exchange mathematical ideas over many years. Another of Cayley's friends was Sylvester who was also in the legal profession. The two both worked at the courts of Lincoln's Inn in London and they discussed deep mathematical questions during their working day. During these 14 years as a lawyer Cayley published about 250 mathematical papers - how many full time mathematicians could compare with the productivity of this 'amateur'.

In 1863 Cayley was appointed Sadleirian professor of Pure Mathematics at Cambridge. This involved a very large decrease in income for Cayley who now had to manage on a salary only a fraction of that which he had earned as a skilled lawyer. However Cayley was very happy to have the chance to devote himself entirely to mathematics.

As Sadleirian professor of Pure Mathematics his duties were

to explain and teach the principles of pure mathematics and to apply himself to the advancement of that science.

Cayley was to more than fulfil these conditions. He published over 900 papers and notes covering nearly every aspect of modern mathematics. The most important of his work is in developing the algebra of matrices, work in non-euclidean geometry and n-dimensional geometry.

As early as 1849 Cayley a paper linking his ideas on permutations with Cauchy's. In 1854 Cayley wrote two papers which are remarkable for the insight they have of abstract groups. At that time the only known groups were groups of permutations and even this was a radically new area, yet Cayley defines an abstract group and gives a table to display the group multiplication. He gives the 'Cayley tables' of some special permutation groups but, much more significantly for the introduction of the abstract group concept, he realised that matrices and quaternions were groups.

Cayley developed the theory of algebraic invariance, and his development of n-dimensional geometry has been applied in physics to the study of the space-time continuum. His work on matrices served as a foundation for quantum mechanics, which was developed by Werner Heisenberg in 1925. Cayley also suggested that euclidean and non-euclidean geometry are special types of geometry. He united projective geometry and metrical geometry which is dependent on sizes of angles and lengths of lines.

In 1881 he was invited to give a course of lectures at Johns Hopkins University in the USA, where his friend Sylvester was professor of mathematics. He spent January to May in 1882 at Johns Hopkins University where he lectured on Abelian and Theta Functions.

In 1883 Cayley became President of the British Association for the Advancement of Science. In his presidential address Cayley gave an elementary account of his own views of mathematics. His views of geometry were

It is well known that Euclid's twelfth axiom, even in Playfair's form of it, has been considered as needing demonstration: and that Lobachevsky constructed a perfectly consistent theory, wherein this axiom was assumed not to hold good, or say a system of non-Euclidean plane geometry. My own view is that Euclid's twelfth axiom in Playfair's form of it does not need demonstration, but is part of our experience - the space, that is, which we become acquainted with by experience, but which is the representation lying at the foundation of all external experience. Riemann's view ... is that, having in intellectu a more general notion of space (in fact a notion of non-Euclidean space), we learn by experience that space (the physical space of our experience) is, if not exactly, at least to the highest degree of approximation, Euclidean space. But suppose the physical space of our experience to be thus only approximately Euclidean space, what is the consequence which follows? Not that the propositions of geometry are only approximately true, but that they remain absolutely true in regard to that Euclidean space which has been so long regarded as being the physical space of our experience.

Two descriptions of Cayley, both of him as an old man, are interesting. Macfarlane [13] says

... I attended a meeting of the Mathematical Society of London. The room was small, and some twelve mathematicians were assembled round a table, among them was Prof. Cayley ... At the close of the meeting Cayley gave me a cordial handshake and referred in the kindest terms to my papers which he had read. He was then about 60 years old, considerably bent, and not filling his clothes. What was most remarkable about him was the active glance of his grey eyes and his peculiar boyish smile.

... a thin weak-looking individual with a large head and face marked with small-pox: he speaks with difficulty and stutters slightly. He never sits upright on his chair but with his posterior on the very edge he leans one elbow on the seat of the chair and throws the other arm over the back.